Can we express the spectral norm of a matrix $F$ in terms of the singular values of the sub-matrix $K$?

Question

Let $\bf K$ be a generic $n \times n$ matrix, and let

$$ {\bf F} := \begin{bmatrix} 0 & {\bf 1}_n^\top \\ {\bf 1}_n & {\bf K} \end{bmatrix} $$

Can we express the spectral norm of $\bf F$ in terms of singular values of $\bf K$ for this specific matrix? In particular, can we write such an equation?

$$ \lVert {\bf F} \rVert = f(\sigma({\bf K})) $$